Cayley transform for Toeplitz and dual matrices

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-10-09 DOI:10.1016/j.laa.2024.10.007

Tikesh Verma , Debasisha Mishra , Michael Tsatsomeros

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引用次数: 0

Abstract

Let an

n \times n

complex matrix A be such that

I + A

is invertible. The Cayley transform of A, denoted by

C (A)

, is defined as

C (A) = {(I + A)}^{- 1} (I - A) .

Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.

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托普利兹矩阵和对偶矩阵的 Cayley 变换

设 n×n 复矩阵 A 的 I+A 是可逆矩阵。A 的 Cayley 变换（用 C(A) 表示）定义为：C(A)=(I+A)-1(I-A)。Fallat 和 Tsatsomeros（2002）[5] 以及 Mondal 等人（2024）[15] 在 P 矩阵、H 矩阵、M 矩阵、全正矩阵、正定矩阵、几乎偏赫米特矩阵和半正定矩阵的背景下研究了矩阵 A 的 Cayley 变换。本文将继续研究托普利兹矩阵、循环矩阵、单能矩阵和对偶矩阵的 Cayley 变换。本文建立了对偶矩阵的 Cayley 变换表达式。证明了对偶对称矩阵的 Cayley 变换总是对偶对称矩阵。讨论了对偶倾斜对称矩阵的 Cayley 变换。并用实例对结果进行了说明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.